740 BELL SYSTEM TECHNICAL JOURNAL 



taking for c the spacing between consecutive lines of atoms in the 

 surface-layer of the nickel crystal which diffracted the electrons, where 

 Fraunhofer had taken the spacing between the wires of the grid which 

 was his primitive grating. 



Next it is important to discover how distinct these maxima are; 

 whether, when the amplitude of the vibration in the focal plane is 

 plotted against 6, the peaks are broad and fiattish or narrow and sharp. 

 This too can be foretold without the labour of a complete solution of 

 the problem. Taking any of the principal maxima — say, that of nth 

 order, which is located at the angle Qn = arc sin (wX/c) — let us inquire 

 how near to it the amplitude will sink to zero. 



Now the nth. of the principal maxima is located by the condition 

 that the phase of the contribution of every slit is 2mr in arrear of that 

 of the slit preceding. If there are 2M slits altogether (it is con- 

 venient to suppose the total number to be even, though whether 

 it is even or odd makes no appreciable difference), then at 0„ the 

 contribution of the last slit is (2M — \)n-2-K behind that of the 

 first. Estimate now the vibration at the point in the focal plane 

 — call its direction-angle (0„ + A)^ — where the contribution of the last 

 sht is {2M - l)(w -f l/2M)27r behind that of the first. It is readily 

 shown that here the component vibration due to the last or 2Mth slit 

 is exactly equal in magnitude and opposite in phase to that which is 

 produced by the ilfth of the slits; and in the same way every ruling 

 of one-half of the grating may be paired off with the corresponding 

 ruling of the other half, their effects destroying each other pair by pair. 

 At the angle {On + A), therefore, there is darkness; and likewise at 

 the angle {On — A'), where in the focal plane or the infinitely distant 

 plane the contributions from the first and the last slit arrive with a 



phase-difference of (2ilf — 1) I w — Trjr ) 27r. 



The entire peak culminating in the wth principal maximum is 

 consequently bounded by the directions (0„ — A') and {Qn + A); and 

 it is easy to see that the greater the number of rulings (the spacing 

 being supposed to remain the same) the narrower and sharper is the 

 peak, and the more accurately can the location of its summit and there- 

 fore the wave-length be determined. Its breadth, in fact, varies 

 inversely as the number of rulings or "lines." This is shown by 

 writing down the formulae for the angles corresponding to the minima 

 which bound it. We have: 



sin (S. + A) = („ +^)\: sin (». - A') = {n - ±-^\. 



